Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Log returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of gross returns

##       vmr             vhr             pmr             phr       
##  Min.   :0.868   Min.   :0.849   Min.   :0.904   Min.   :0.878  
##  1st Qu.:1.044   1st Qu.:1.039   1st Qu.:1.042   1st Qu.:1.068  
##  Median :1.097   Median :1.099   Median :1.084   Median :1.128  
##  Mean   :1.070   Mean   :1.085   Mean   :1.065   Mean   :1.095  
##  3rd Qu.:1.136   3rd Qu.:1.160   3rd Qu.:1.107   3rd Qu.:1.182  
##  Max.   :1.168   Max.   :1.214   Max.   :1.141   Max.   :1.208  
##       mmr             mhr       
##  Min.   :0.988   Min.   :0.977  
##  1st Qu.:1.013   1st Qu.:1.013  
##  Median :1.085   Median :1.113  
##  Mean   :1.066   Mean   :1.087  
##  3rd Qu.:1.101   3rd Qu.:1.128  
##  Max.   :1.133   Max.   :1.207
##       vmrl      
##  Min.   :0.801  
##  1st Qu.:1.013  
##  Median :1.085  
##  Mean   :1.061  
##  3rd Qu.:1.128  
##  Max.   :1.193
##            vmr   vhr   pmr   phr   mmr   mhr
## Min.   : 0.868 0.849 0.904 0.878 0.988 0.977
## 1st Qu.: 1.044 1.039 1.042 1.068 1.013 1.013
## Median : 1.097 1.099 1.084 1.128 1.085 1.113
## Mean   : 1.070 1.085 1.065 1.095 1.066 1.087
## 3rd Qu.: 1.136 1.160 1.107 1.182 1.101 1.128
## Max.   : 1.168 1.214 1.141 1.208 1.133 1.207

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.095 phr 1.182 phr 1.214 vhr
0.977 mhr 1.044 vmr 1.113 mhr 1.087 mhr 1.160 vhr 1.208 phr
0.904 pmr 1.042 pmr 1.099 vhr 1.085 vhr 1.136 vmr 1.207 mhr
0.878 phr 1.039 vhr 1.097 vmr 1.070 vmr 1.128 mhr 1.168 vmr
0.868 vmr 1.013 mmr 1.085 mmr 1.066 mmr 1.107 pmr 1.141 pmr
0.849 vhr 1.013 mhr 1.084 pmr 1.065 pmr 1.101 mmr 1.133 mmr

Covariance

## cov(vmr, pmr) =  -0.001094875
## cov(vhr, phr) =  -0.0001730651

Velliv medium risk, 2011 - 2023

## 
## AIC: -27.8497 
## BIC: -25.58991 
## m: 0.0480931 
## s: 0.1198426 
## nu (df): 3.303595 
## xi: 0.03361192 
## R^2: 0.993 
## 
## An R^2 of 0.993 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 63.16667 percent
## What is the chance of gaining min 25 %? >= 49.33333 percent
## What is the chance of gaining min 50 %? >= 40.16667 percent
## What is the chance of gaining min 90 %? >= 32.66667 percent
## What is the chance of gaining min 99 %? >= 31.5 percent

QQ Plot

The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 278.995 kr.
## SD of portfolio index value after 20 years: 123.583 kr.
## Min total portfolio index value after 20 years: 9.812 kr.
## Max total portfolio index value after 20 years: 929.858 kr.
## 
## Share of paths finishing below 100: 4.72 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.1724769 0.3205692

Objective function plots

Velliv medium risk, 2007 - 2023

Fit to skew t distribution

## 
## AIC: -34.35752 
## BIC: -31.02467 
## m: 0.05171176 
## s: 0.1149408 
## nu (df): 2.706099 
## xi: 0.5049945 
## R^2: 0.978 
## 
## An R^2 of 0.978 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 58.66667 percent
## What is the chance of gaining min 25 %? >= 47.5 percent
## What is the chance of gaining min 50 %? >= 40.16667 percent
## What is the chance of gaining min 90 %? >= 34 percent
## What is the chance of gaining min 99 %? >= 33 percent

QQ Plot

The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 293.384 kr.
## SD of portfolio index value after 20 years: 118.147 kr.
## Min total portfolio index value after 20 years: 0.785 kr.
## Max total portfolio index value after 20 years: 2029.133 kr.
## 
## Share of paths finishing below 100: 3.11 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.1842753 0.3193925

Objective function plots

Velliv high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -21.42488 
## BIC: -19.16508 
## m: 0.06471454 
## s: 0.1499924 
## nu (df): 3.144355 
## xi: 0.002367034 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 64.66667 percent
## What is the chance of gaining min 25 %? >= 47.83333 percent
## What is the chance of gaining min 50 %? >= 36.83333 percent
## What is the chance of gaining min 90 %? >= 28 percent
## What is the chance of gaining min 99 %? >= 26.5 percent

QQ Plot

The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 404.855 kr.
## SD of portfolio index value after 20 years: 217.274 kr.
## Min total portfolio index value after 20 years: 0.01 kr.
## Max total portfolio index value after 20 years: 1713.743 kr.
## 
## Share of paths finishing below 100: 4.3 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.5302163 0.4155546

Objective function plots

PFA medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -33.22998 
## BIC: -30.97018 
## m: 0.05789224 
## s: 0.1234592 
## nu (df): 2.265273 
## xi: 0.477324 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 52.83333 percent
## What is the chance of gaining min 25 %? >= 44 percent
## What is the chance of gaining min 50 %? >= 38.83333 percent
## What is the chance of gaining min 90 %? >= 34.66667 percent
## What is the chance of gaining min 99 %? >= 34 percent

QQ Plot

The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.

##  [1] -0.091256521 -0.003731241  0.027312079  0.045808232  0.059068633
##  [6]  0.069575113  0.078454727  0.086316936  0.093536451  0.100370932
## [11]  0.107018607  0.114081432  0.127604387

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 321.882 kr.
## SD of portfolio index value after 20 years: 106.531 kr.
## Min total portfolio index value after 20 years: 0.01 kr.
## Max total portfolio index value after 20 years: 1121.574 kr.
## 
## Share of paths finishing below 100: 1.94 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.2338345 0.2992717

Objective function plots

PFA high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -23.72565 
## BIC: -21.46585 
## m: 0.08386034 
## s: 0.1210107 
## nu (df): 3.184569 
## xi: 0.01790306 
## R^2: 0.964 
## 
## An R^2 of 0.964 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 56.83333 percent
## What is the chance of gaining min 25 %? >= 43.16667 percent
## What is the chance of gaining min 50 %? >= 34.16667 percent
## What is the chance of gaining min 90 %? >= 26.83333 percent
## What is the chance of gaining min 99 %? >= 25.66667 percent

QQ Plot

The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 555.222 kr.
## SD of portfolio index value after 20 years: 244.751 kr.
## Min total portfolio index value after 20 years: 0.765 kr.
## Max total portfolio index value after 20 years: 1828.26 kr.
## 
## Share of paths finishing below 100: 0.95 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.7617723 0.4255421

Objective function plots

Mix medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -36.9603 
## BIC: -34.7005 
## m: 0.05902873 
## s: 0.08757749 
## nu (df): 2.772621 
## xi: 0.02904471 
## R^2: 0.89 
## 
## An R^2 of 0.89 suggests that the fit is not completely random.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 53.16667 percent
## What is the chance of gaining min 25 %? >= 44.16667 percent
## What is the chance of gaining min 50 %? >= 38.66667 percent
## What is the chance of gaining min 90 %? >= 34.16667 percent
## What is the chance of gaining min 99 %? >= 33.5 percent

QQ Plot

The fit suggests big losses for the lowest percentiles, which are not present in the data.
So the fit is actually a very cautious estimate.

Data vs fit

Let’s plot the fit and the observed returns together.

Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 324.055 kr.
## SD of portfolio index value after 20 years: 98.254 kr.
## Min total portfolio index value after 20 years: 3.973 kr.
## Max total portfolio index value after 20 years: 723.752 kr.
## 
## Share of paths finishing below 100: 1.13 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 300.815 kr.
## SD of portfolio index value after 20 years: 82.422 kr.
## Min total portfolio index value after 20 years: 42.661 kr.
## Max total portfolio index value after 20 years: 1008.214 kr.
## 
## Share of paths finishing below 100: 0.34 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.1398383 0.2595942

Objective function plots

Mix high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -24.26084 
## BIC: -22.00104 
## m: 0.0822419 
## s: 0.07129843 
## nu (df): 89.86289 
## xi: 0.7697502 
## R^2: 0.961 
## 
## An R^2 of 0.961 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 52.5 percent
## What is the chance of gaining min 25 %? >= 45 percent
## What is the chance of gaining min 50 %? >= 38.33333 percent
## What is the chance of gaining min 90 %? >= 31.16667 percent
## What is the chance of gaining min 99 %? >= 29.83333 percent

QQ Plot

The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that the high risk mix provides a much better upside and smaller downside.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 498.653 kr.
## SD of portfolio index value after 20 years: 155.916 kr.
## Min total portfolio index value after 20 years: 150.86 kr.
## Max total portfolio index value after 20 years: 1463.716 kr.
## 
## Share of paths finishing below 100: 0 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 479.218 kr.
## SD of portfolio index value after 20 years: 164.197 kr.
## Min total portfolio index value after 20 years: 55.397 kr.
## Max total portfolio index value after 20 years: 1335.395 kr.
## 
## Share of paths finishing below 100: 0.12 percent

Many simulations

1e6 paths:

# Down-and-out simulation:
# Probability of down-and-out: 0 percent
# 
# Mean portfolio index value after 20 years: 478.339 kr.
# SD of portfolio index value after 20 years: 163.093 kr.
# Min total portfolio index value after 20 years: 2.233 kr.
# Max total portfolio index value after 20 years: 1561.965 kr.
# 
# Share of paths finishing below 100: 0.1181 percent

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.5927304 0.3361558

Objective function plots

Compare pension plans

Risk of max loss

Risk of max loss of x percent for a single period (year).
x values are row names.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m mix_h
0 21.167 17.833 19.667 11.833 14.000 12.333 12.667
5 12.167 9.333 12.500 5.667 8.333 5.833 3.833
10 7.000 5.000 8.000 3.000 5.000 2.833 0.500
25 1.333 0.833 2.167 0.500 1.000 0.333 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Worst ranking for loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.167 Velliv_m 12.500 Velliv_h 8.000 Velliv_h 2.167 Velliv_h 0 Velliv_m 0 Velliv_m 0 Velliv_m
19.667 Velliv_h 12.167 Velliv_m 7.000 Velliv_m 1.333 Velliv_m 0 Velliv_m_l 0 Velliv_m_l 0 Velliv_m_l
17.833 Velliv_m_l 9.333 Velliv_m_l 5.000 Velliv_m_l 1.000 PFA_h 0 Velliv_h 0 Velliv_h 0 Velliv_h
14.000 PFA_h 8.333 PFA_h 5.000 PFA_h 0.833 Velliv_m_l 0 PFA_m 0 PFA_m 0 PFA_m
12.667 mix_h 5.833 mix_m 3.000 PFA_m 0.500 PFA_m 0 PFA_h 0 PFA_h 0 PFA_h
12.333 mix_m 5.667 PFA_m 2.833 mix_m 0.333 mix_m 0 mix_m 0 mix_m 0 mix_m
11.833 PFA_m 3.833 mix_h 0.500 mix_h 0.000 mix_h 0 mix_h 0 mix_h 0 mix_h

Chance of min gains

Chance of min gains of x percent for a single period (year).
x values are row names.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m mix_h
0 78.833 82.167 80.333 88.167 86.000 87.667 87.333
5 63.833 65.000 69.333 71.667 76.000 71.667 70.167
10 40.833 36.000 53.333 32.500 59.667 35.500 46.000
25 0.000 0.000 0.000 0.000 0.000 0.000 0.833
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Best ranking for gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
88.167 PFA_m 76.000 PFA_h 59.667 PFA_h 0.833 mix_h 0 Velliv_m 0 Velliv_m
87.667 mix_m 71.667 PFA_m 53.333 Velliv_h 0.000 Velliv_m 0 Velliv_m_l 0 Velliv_m_l
87.333 mix_h 71.667 mix_m 46.000 mix_h 0.000 Velliv_m_l 0 Velliv_h 0 Velliv_h
86.000 PFA_h 70.167 mix_h 40.833 Velliv_m 0.000 Velliv_h 0 PFA_m 0 PFA_m
82.167 Velliv_m_l 69.333 Velliv_h 36.000 Velliv_m_l 0.000 PFA_m 0 PFA_h 0 PFA_h
80.333 Velliv_h 65.000 Velliv_m_l 35.500 mix_m 0.000 PFA_h 0 mix_m 0 mix_m
78.833 Velliv_m 63.833 Velliv_m 32.500 PFA_m 0.000 mix_m 0 mix_h 0 mix_h

MC risk percentiles

Risk of loss from first to last period.

_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

_m is medium.
_h is high.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 4.72 3.11 4.30 1.94 0.95 1.13 0 0.34 0.12
5 4.14 2.78 3.82 1.81 0.81 1.02 0 0.23 0.09
10 3.72 2.27 3.35 1.70 0.70 0.90 0 0.13 0.09
25 2.42 1.52 2.44 1.23 0.47 0.55 0 0.07 0.03
50 0.86 0.69 1.11 0.59 0.25 0.27 0 0.01 0.00
90 0.01 0.11 0.10 0.18 0.04 0.02 0 0.00 0.00
99 0.00 0.01 0.01 0.06 0.01 0.00 0 0.00 0.00

1e6 simulation paths of mhr_b:

0 5 10 25 50 90 99
prob_pct 0.118 0.095 0.076 0.036 0.008 0 0

Worst ranking for MC loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.72 Velliv_m 4.14 Velliv_m 3.72 Velliv_m 2.44 Velliv_h 1.11 Velliv_h 0.18 PFA_m 0.06 PFA_m
4.30 Velliv_h 3.82 Velliv_h 3.35 Velliv_h 2.42 Velliv_m 0.86 Velliv_m 0.11 Velliv_m_l 0.01 Velliv_m_l
3.11 Velliv_m_l 2.78 Velliv_m_l 2.27 Velliv_m_l 1.52 Velliv_m_l 0.69 Velliv_m_l 0.10 Velliv_h 0.01 Velliv_h
1.94 PFA_m 1.81 PFA_m 1.70 PFA_m 1.23 PFA_m 0.59 PFA_m 0.04 PFA_h 0.01 PFA_h
1.13 mix_m_a 1.02 mix_m_a 0.90 mix_m_a 0.55 mix_m_a 0.27 mix_m_a 0.02 mix_m_a 0.00 Velliv_m
0.95 PFA_h 0.81 PFA_h 0.70 PFA_h 0.47 PFA_h 0.25 PFA_h 0.01 Velliv_m 0.00 mix_m_a
0.34 mix_m_b 0.23 mix_m_b 0.13 mix_m_b 0.07 mix_m_b 0.01 mix_m_b 0.00 mix_h_a 0.00 mix_h_a
0.12 mix_h_b 0.09 mix_h_b 0.09 mix_h_b 0.03 mix_h_b 0.00 mix_h_a 0.00 mix_m_b 0.00 mix_m_b
0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_b 0.00 mix_h_b 0.00 mix_h_b

MC gains percentiles

Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 95.28 96.89 95.70 98.06 99.05 98.87 100.00 99.66 99.88
5 94.67 96.40 95.17 97.80 98.93 98.72 100.00 99.55 99.83
10 94.00 95.89 94.73 97.56 98.78 98.54 100.00 99.39 99.82
25 91.57 93.96 93.21 96.68 98.42 97.85 100.00 99.01 99.72
50 85.87 89.95 90.33 94.73 97.50 96.23 100.00 97.42 99.31
100 71.62 78.11 83.11 87.55 94.60 89.72 99.66 89.73 97.46
200 39.14 44.61 64.53 58.58 85.28 59.42 93.18 48.34 87.18
300 16.10 18.01 44.83 22.36 71.62 22.28 71.24 11.58 65.20
400 5.20 4.56 29.08 4.45 54.84 3.77 43.90 1.27 42.17
500 1.45 1.16 17.33 0.59 39.30 0.22 22.71 0.07 22.12
1000 0.00 0.03 0.59 0.01 2.28 0.00 0.25 0.00 0.11

1e6 simulation paths of mhr_b:

0 5 10 25 50 100 200 300 400 500 1000
prob 99.882 99.854 99.824 99.686 99.301 97.513 86.912 65.992 41.486 21.693 0.086

Best ranking for MC gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 mix_h_a 100.00 mix_h_a 100.00 mix_h_a 100.00 mix_h_a 100.00 mix_h_a 99.66 mix_h_a
99.88 mix_h_b 99.83 mix_h_b 99.82 mix_h_b 99.72 mix_h_b 99.31 mix_h_b 97.46 mix_h_b
99.66 mix_m_b 99.55 mix_m_b 99.39 mix_m_b 99.01 mix_m_b 97.50 PFA_h 94.60 PFA_h
99.05 PFA_h 98.93 PFA_h 98.78 PFA_h 98.42 PFA_h 97.42 mix_m_b 89.73 mix_m_b
98.87 mix_m_a 98.72 mix_m_a 98.54 mix_m_a 97.85 mix_m_a 96.23 mix_m_a 89.72 mix_m_a
98.06 PFA_m 97.80 PFA_m 97.56 PFA_m 96.68 PFA_m 94.73 PFA_m 87.55 PFA_m
96.89 Velliv_m_l 96.40 Velliv_m_l 95.89 Velliv_m_l 93.96 Velliv_m_l 90.33 Velliv_h 83.11 Velliv_h
95.70 Velliv_h 95.17 Velliv_h 94.73 Velliv_h 93.21 Velliv_h 89.95 Velliv_m_l 78.11 Velliv_m_l
95.28 Velliv_m 94.67 Velliv_m 94.00 Velliv_m 91.57 Velliv_m 85.87 Velliv_m 71.62 Velliv_m
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
93.18 mix_h_a 71.62 PFA_h 54.84 PFA_h 39.30 PFA_h 2.28 PFA_h
87.18 mix_h_b 71.24 mix_h_a 43.90 mix_h_a 22.71 mix_h_a 0.59 Velliv_h
85.28 PFA_h 65.20 mix_h_b 42.17 mix_h_b 22.12 mix_h_b 0.25 mix_h_a
64.53 Velliv_h 44.83 Velliv_h 29.08 Velliv_h 17.33 Velliv_h 0.11 mix_h_b
59.42 mix_m_a 22.36 PFA_m 5.20 Velliv_m 1.45 Velliv_m 0.03 Velliv_m_l
58.58 PFA_m 22.28 mix_m_a 4.56 Velliv_m_l 1.16 Velliv_m_l 0.01 PFA_m
48.34 mix_m_b 18.01 Velliv_m_l 4.45 PFA_m 0.59 PFA_m 0.00 Velliv_m
44.61 Velliv_m_l 16.10 Velliv_m 3.77 mix_m_a 0.22 mix_m_a 0.00 mix_m_a
39.14 Velliv_m 11.58 mix_m_b 1.27 mix_m_b 0.07 mix_m_b 0.00 mix_m_b

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
m 0.048 0.052 0.065 0.058 0.084 0.059 0.082
s 0.120 0.115 0.150 0.123 0.121 0.088 0.071
nu 3.304 2.706 3.144 2.265 3.185 2.773 89.863
xi 0.034 0.505 0.002 0.477 0.018 0.029 0.770
R-squared 0.993 0.978 0.991 0.991 0.964 0.890 0.961

Fit statistics ranking

m ranking s ranking R-squared ranking
0.084 PFA_high 0.071 mix_high 0.993 Velliv_medium
0.082 mix_high 0.088 mix_medium 0.991 Velliv_high
0.065 Velliv_high 0.115 Velliv_medium_long 0.991 PFA_medium
0.059 mix_medium 0.120 Velliv_medium 0.978 Velliv_medium_long
0.058 PFA_medium 0.121 PFA_high 0.964 PFA_high
0.052 Velliv_medium_long 0.123 PFA_medium 0.961 mix_high
0.048 Velliv_medium 0.150 Velliv_high 0.890 mix_medium

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_m_b mix_h_a mix_h_b
mc_m 278.995 293.384 404.855 321.882 555.222 324.055 300.815 498.653 479.218
mc_s 123.583 118.147 217.274 106.531 244.751 98.254 82.422 155.916 164.197
mc_min 9.812 0.785 0.010 0.010 0.765 3.973 42.661 150.860 55.397
mc_max 929.858 2029.133 1713.743 1121.574 1828.260 723.752 1008.214 1463.716 1335.395
dao_pct 0.000 0.000 0.010 0.010 0.000 0.000 0.000 0.000 0.000
losing_pct 4.720 3.110 4.300 1.940 0.950 1.130 0.340 0.000 0.120

Ranking

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking losing_pct ranking
555.222 PFA_h 82.422 mix_m_b 150.860 mix_h_a 2029.133 Velliv_m_l 0.00 Velliv_m 0.00 mix_h_a
498.653 mix_h_a 98.254 mix_m_a 55.397 mix_h_b 1828.260 PFA_h 0.00 Velliv_m_l 0.12 mix_h_b
479.218 mix_h_b 106.531 PFA_m 42.661 mix_m_b 1713.743 Velliv_h 0.00 PFA_h 0.34 mix_m_b
404.855 Velliv_h 118.147 Velliv_m_l 9.812 Velliv_m 1463.716 mix_h_a 0.00 mix_m_a 0.95 PFA_h
324.055 mix_m_a 123.583 Velliv_m 3.973 mix_m_a 1335.395 mix_h_b 0.00 mix_m_b 1.13 mix_m_a
321.882 PFA_m 155.916 mix_h_a 0.785 Velliv_m_l 1121.574 PFA_m 0.00 mix_h_a 1.94 PFA_m
300.815 mix_m_b 164.197 mix_h_b 0.765 PFA_h 1008.214 mix_m_b 0.00 mix_h_b 3.11 Velliv_m_l
293.384 Velliv_m_l 217.274 Velliv_h 0.010 Velliv_h 929.858 Velliv_m 0.01 Velliv_h 4.30 Velliv_h
278.995 Velliv_m 244.751 PFA_h 0.010 PFA_m 723.752 mix_m_a 0.01 PFA_m 4.72 Velliv_m

Compare Gaussian and skewed t-distribution fits

Gaussian fits

vmr vhr pmr phr mmr mhr
m 0.064 0.077 0.061 0.085 0.062 0.081
s 0.081 0.099 0.063 0.101 0.048 0.070

Gaussian QQ plots

Gaussian vs skewed t

Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:

vmr vhr pmr phr mmr mhr
P_norm(X_min) 0.571 0.758 0.511 1.676 5.971 6.842
P_norm(X_max) 13.230 11.876 12.922 15.359 9.628 6.429
P_t(X_min) 5.377 5.457 3.489 4.315 10.570 8.015
P_t(X_max) 0.118 0.001 2.825 0.188 0.488 5.141

Average number of years between min or max events (respectively):

vmr vhr pmr phr mmr mhr
norm: avg yrs btw min 175.248 131.911 195.568 59.669 16.748 14.616
norm: avg yrs btw max 7.559 8.420 7.739 6.511 10.386 15.556
t: avg yrs btw min 18.596 18.324 28.663 23.173 9.461 12.476
t: avg yrs btw max 848.548 178349.076 35.400 531.552 205.104 19.450

Comments

(Ignoring mhr_a…)

mhr has some nice properties:
- It has a relatively high nu value of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds have nu values close to 3, except phr which is even worse at close to 2. (Note that for a Gaussian, nu is infinite.)
- It has the lowest losing percentage of all simulations, which is better than 1/6 that of phr.
- It has a DAO percentage of 0, which is the same as mmr, and less than phr.
- Only phr has a higher mc_m.
- It has a smaller mc_s than the individual components, vhr and phr.
- It has the highest xi of all fits, suggesting less left skewness. Density plots for vmr, phr and mmr have an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely low xi values. The density plot for mhr is by far the most symmetrical of all the fits. As seen in the section “Compare Gaussian and skewed t-distribution fits”, the other skewed t-distribution fits don’t capture the max observed returns at all.
- Only mmr has as higher mc_min. However, that of mmr is 18 times higher with 62, so mmr is a clear winner here.
- Naturally, it has a mc_max smaller than the individual components, vhr and phr, but ca. 1.5 times higher then mmr.
- All the first 4 moments converge nicely. For all other fits, the 4th moment doesn’t seem to converge.

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”

Appendix

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): -0.008375401 
## s(data_x): 0.4184349 
## m(data_y): 9.445322 
## s(data_y): 2.665942 
## 
## m(data_x + data_y): 4.718473 
## s(data_x + data_y): 1.429784

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
94.308 94.032 5.930 6.447
94.046 94.419 6.020 6.583
94.537 94.423 6.064 6.567
94.585 94.491 6.202 6.386
94.470 94.184 6.192 6.242
94.480 94.510 6.141 6.366
94.484 94.503 5.843 6.303
94.501 94.179 5.840 6.397
94.314 94.193 5.941 6.350
94.222 94.482 6.180 6.661 
##       m_a             m_b             s_a             s_b       
##  Min.   :94.05   Min.   :94.03   Min.   :5.840   Min.   :6.242  
##  1st Qu.:94.31   1st Qu.:94.19   1st Qu.:5.933   1st Qu.:6.354  
##  Median :94.48   Median :94.42   Median :6.042   Median :6.391  
##  Mean   :94.39   Mean   :94.34   Mean   :6.035   Mean   :6.430  
##  3rd Qu.:94.50   3rd Qu.:94.49   3rd Qu.:6.170   3rd Qu.:6.537  
##  Max.   :94.59   Max.   :94.51   Max.   :6.202   Max.   :6.661

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05940   Min.   :0.04936  
##  1st Qu.:0.06631   1st Qu.:0.05986  
##  Median :0.07033   Median :0.06726  
##  Mean   :0.07071   Mean   :0.06858  
##  3rd Qu.:0.07282   3rd Qu.:0.07670  
##  Max.   :0.08455   Max.   :0.09120

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192